CHANNEL ESTIMATION FOR SPACE-TIME ORTHOGONAL BLOCK CODES. Cristian Budianu and Lang Tong

Transcription

1 A ) CHANNEL ESTIMATION FOR SPACE-TIME ORTHOGONAL BLOCK CODES Crisian Budianu and Lang Tong School of Elecrical Engineering, Cornell Universiy, Ihaca, NY cris@eecornelledu, long@eecornelledu ABSTRACT Space-ie coding is an efficien echnique o exploi ransiing and receiving diversiies for wireless channels One of he key coponens in he design of receivers for space-ie codes is channel esiaion For he block codes based on orhogonal design proposed Alaoui and Tarokh, a new class of pilo assised channel esiaion schees are presened where pilo sybols are superiposed on he daa sybols Placeen sraegies are exained for heir efficiency in channel esiaion and daa ransission 1 INTRODUCTION A ajor challenge in space-ie counicaions in a wireless environen is channel esiaion Typically, pilo sybols are necessary o acquire he channel [1] The inserion of pilo sybols, however, ay induce unaccepable overhead and lower he daa hroughpu Here syse designers us consider wo conradicory goals On he one hand, i is desirable o iniize he nuber of pilo sybols in a daa packe so ha ore inforaion carrying sybols can be ransied On he oher hand, ore pilo sybols resul in beer channel esiaion hence reducing sybol error rae and he need for packe reransissions In his paper, we consider he channel esiaion proble in he fraework of space-ie block codes fro orhogonal designs proposed Tarokh e al [] Our goal is o exaine he effecs of he nuber, power and placeen of pilo sybols in daa packes This radeoff can be exained he Craér-Rao Lower Bound (CRLB) and uual inforaion Boh quaniies are funcions of he nuber of pilo sybols insered in he daa packe and he locaions of hese known sybols Specifically, we consider he possibiliy of superiposing pilo sybols on daa sreas This enables coninuous ransission of inforaion sybols while channel esiaion and racking can be perfored siulaneously I also offers he flexibiliy o allocae power dynaically for channel esiaion and sybol ransission We show ha cerain placeens of pilos for he orhogonal block codes are equivalen in he sense ha hey have he idenical Fisher inforaion arices, which iplies hey have he sae CRLBs The proble of channel esiaion for space-ie code was considered Naguib e al [1] By using orhogonal pilo sybols, a siple channel esiaion algorih was proposed ha This work was suppored in par he Mulidisciplinary Universiy Research Iniiaive (MURI) under he Office of Naval Research Conrac N and he Ary Research Office under Gran ARO- DAAB used only observaions associaed wih pilo sybols By ignoring observaions fro daa sybols, such echniques do no lead o efficien esiaion The locaion of pilo sybols was also fixed in [1] The idea of using superiposed pilo for channel esiaion has been proposed Makrakis and Feher [6], Hoeher and Tufvesson [4], Manon and Hua [5], Ohno and Giannakis[10] The analysis presened in hese papers, however, do no apply direcly o he space-ie block codes This paper is organized as follows In Secion, we presen he fraework and assupions In Subsecion 31, a general schee for pilo sybol placeen is presened We also discuss radeoffs aong key facors In Subsecions 3 and 33 he perforance of differen placeen schees is exained using Craér-Rao Lower Bounds and uual inforaion respecively Siulaions and nuerical resuls are presened in Secion 4 and are followed concluding rearks MODEL AND ASSUMPTIONS Consider a uliple anenna syse shown in Figure 1 where here are ransiers and receivers In his paper we consider only rae one codes and real sybols Sybols are ransied in blocks of size Each block is ransied wihin sybol periods For block, is ransied fro anennas, and he h colun of corresponds o he ransied vecor in he h sybol inerval The space-ie code proposed Tarokh e al [] has he following for % "! (1) where # "!'&(&'&'! ) is he block of N ransied sybols and #* +) is he space-ie block code I is shown in [],[9] ha using single user deecors in parallel, he bes perforance is obained if -,*/ , 498 <; = > > > = 65 () For rae one codes and real sybols i is shown, in he heory of orhogonal designs, ha he faily #: exiss if and only if =, 4 or 8 Under he quasi-saic fla fading odel, he received signal arix for block is AB DC6EF <; GCEF Ḧ! (3)

2 ~ J T I I J K Transier L Transier PSfrag replaceens I I J Receiver L MAN GC6EO Receiver Verical Placeen - ^ : The verical placeen is described ~Š r!vƒ ~ e, L!Vƒ L! &'&(&'! L!'&'&'&'!!Y5-ˆx! &(&'&'! (9) This schee is siilar o he classical raining schees where a block conains only pilo sybols The disadvanage of his schee is ha i is vulnerable o noise burss during he ransission of he raining block Fig 1: An -ransier -receiver space-ie syse where APFQ R is he channel arix and EF is he addiive Gaussian noise In he sequel we need he received signal and he paraeers represened as colun vecors Define We hen have S ÜT vec V!HWX T vec YEF V! (4) Z vec ['A ]\ & (5) S Jba ^`_ c; ëd Z C6Wf"& (6) In his paper, we assue (i) he noise Wf 9gih+Yjk!l] se- are real and n-# )9 L quence is iid; (ii) he sybols 3 CHANNEL ESTIMATION WITH SUPERIMPOSED PILOT SYMBOLS 31 Placeen of Superiposed Pilo Sybols Pilo sybols are usually ebedded in he daa srea for channel esiaions Typically, he paern of pilo placeen is periodic In his paper, we consider he placeen of pilo sybols in cycles of blocks, and we only need o specify he placeen for one cycle We generalize he placeen of pilo sybols allowing superiposed raining In paricular, each ransied sybol consiss of he pilo par o and inforaion carrying par p 1q L 7sr ẗo GCq r ẗp "! (7) where power is allocaed via r Ẍvu wx! L(y Here we assue p is zero ean, uni variance, and o zf#*{ L ) The } z placeen of pilo sybols is equivalen o he specificaion of he power allocaion arix ~6u ~, y, where ~, r 5,ƒ 4!5 L! &'&'&(! Fro his general faily, we ll consider hree special cases shown in Fig Horizonal Placeen- ^ : The horizonal placeen is described, r!vƒr5 L! &'&'&(! ~f, L!Vƒ ˆx! &'&(&'!!5 L!'& &'&'! & (8) The horizonal placeen has he advanage ha all he blocks wihin a cycle have he sae srucure, ha siplifies esiaion a recepion and allows ore efficien racking for ie-varying channels Oher advanages of his schee will be showed laer Unifor placeen - : The unifor placeen schee is described (10) % % (a) Horizonal (c) Unifor ~ e, r!eƒ!5- L! &'&'&'! % (b) Verical PSfrag replaceens (a)horizonal (b)verical (c)unifor Fig : Special placeen schees; he black regions represen he raining sybols The choice of ~ affecs boh CRLB of he channel esiaes and he aoun of inforaion ha can be ransied hrough he channel 3 Craér-Rao Lower Bound (CRLB) The CRLB provides he lower bound on he ean square error (MSE) for all unbiased esiaors The raio beween he MSE of an esiaor and he CRLB easures he efficiency of he algorih in uilizing he inforaion available in he observaion In general, CRLB is a funcion of of he channel arix and he sybol placeen schee Therefore, differen designs of he power allocaion arix lead o differen CRLB Z We consider esiaing he channel vecor using he enire daa record fro one cycle # S L "!(Ž'Ž Ž'! S S u T S L "! Ž'Ž(Ž'! S X ) Define y X& (11) Z The likelihood funcion for he channel paraeer is hen given SX Z h+ SX Z "!" / Z "! (1)

3 J 4 ~ o  Рwhereh+ SX Z "!" / Z is he pdf of he noral disribuion wih ean Z and covariance / Z given J a <; œ Denoe ž K š L 7 ~ Z J a <; K ^ L 7 ~ o / Z diag u Z "!'&'&'&!" Jba Ÿ k Z, and we have - š <; ž ž GC6l L Z (13) = Z y & (14) & (15) Consider ha he channel paraeers are real nubers The eleens of he Fisher inforaion arix (FIM) are given : u Z z : y, 3 H ªs«Z 9 : H ' «C r Z : % 'ª Z : ' O«& (16) Because of he independence of he unknown sybols, arix has a block diagonal srucure, which akes i possible o calculae he FIM as a su of he FIMs for each block Lea 1 The inverse of he arix is given where ² T ³µ ± and 'T 7 c; Vž ± ž DCv² ª ³% ¹ µ ª º ³µ ), we have he fol- Using properies of he orhogonal codes #: lowing heores Theore 1 If }ˆ or» he FIM of he channel paraeers is he sae for he horizonal and verical placeen I follows ha he CRLBs of he paraeers esiaed fro he wo placeen schees are he sae Noe ha he heore above does no hold for N=8 In secion 4 soe nuerical exaples are given Theore Consider only one block, and ˆ or M» Then he FIM of he placeen schee ~ ¼u r ½! &'&(&'! r ½ <¾ ½R< y does no depend on he choice of he uni vecors ( ie, on he indices! À L!'&(&'&! ) or on he BPSK raining sequence used This heore says ha we can change he posiion of he raining sybol in a block of he horizonal placeen wihou affecing he perforance of he schee Again his heore does no hold for MÁ 33 Muual Inforaion Muual inforaion beween he inpu and he oupu easures he efficiency and reliabiliy of daa ransission In [8, 7], Adireddy and Tong exained he effec of pilo sybols on he capaciy of he syse and showed ha placing known sybols judiciously can iprove he ransission rae significanly For space-ie code syses, he placeen of known sybols again play an iporan role If he channel is known, for Gaussian inpu sybols we have SX à MÄÆÅ Ç È ÉRÊ(Ë: cov S 'È 7 ÄÆÅ%ÇÌÈ ÉÍÊ(Ë:¹l 'È Using he properies of he space-ie block codes we have : ÎxÏ ' / Z ^Ml ; Ð ; ¹Ñ ~ ( Cvl Ḧ& (17) Theore 3 The uual inforaion beween he inpu and he oupu does no change under any peruaion of he coefficiens ~ If we consrain he aoun of power ha can be used o ransi pilo signals, ie, = ; = > > > = hen he unifor schee wih ~ " ~ ( M }MÒ"Ó ƒ 4 L &(&'& (!! L &'&'& (18) axiizes he uual inforaion beween he inpu and he oupu 4 SIMULATIONS AND NUMERICAL RESULTS 41 Siulaions The channel esiaion algorih ha was used was a seiblind Maxiu Likelihood Algorih using he scoring ehod The nuerical resuls ha are presened were obained using he following seup The raining sybols were binary, C L! 7 L, he paraeers values were»í! ÔR!G ÀÕ, nuber of blocks considered Z GÖŠMÔµˆ The channel coefficiens were rando nubers OYwR! L(y ƒ Mone Carlo siulaions were perfored We evaluaed he su of he CRLBs for all paraeers 4 Nuerical Resuls Fig3 shows he CRLB and esiaion variance of he ML paraeers vs he power of noise, l In our siulaion, he ML esiaor was iniialized randoly, and i usually converged in less han 10 ieraions We observed ha for reasonable allocaions of power rà wx& Ø, he MSE of he ML esiaor is close o CRLB In Fig4 shows he CRLB vs r for differen powers of noise I can be observed ha he raining schees wih less power allocaed o raining are ore sensiive o noise 1 he following channel was used ÙÛÚ Ü Ý%Þ ß à%áý%þ â(ãµá Ý%Þ â'ä%áeý%þ â â áeý%þ à å%á Ý%Þ à æ%áý%þ ä ä%áeý%þ ß'ãµá Ý%Þ à ä%áeý%þ ß ß%áeÝ%Þ â'ý%áeý%þ ä å%áeý%þ ß à%á Ý%Þ ß ä%þ Ý%Þ à ä(ç

4 In Fig5 he horizonal placeen schee is copared wih he unifor one using he CRLB I can be observed ha he unifor placeen schee has higher CRLB han he horizonal schee, and he difference is significan a high SNR However, a low SNR he wo schees perfor siilarly In Fig6 copares he horizonal and verical placeen for a code wih èá A he curren resoluion he wo placeen schees appear idenical, bu heir FIMs are differen and heir is also a sall difference beween he CRLBs P U γ=08 P U γ=08 variance of ML and 0 CRLB ML CRLB γ=05 ML γ=05 CRLB γ=04 ML γ=04 CRLB γ=06 ML γ= /σ [db] Fig 5: CRLB for differen values of r for he horizonal schee (coninuous line) and unifor schee (dashed line) γ=05 P V P V γ= /σ [db] 15 Fig 3: CRLB (coninous line) and he variance of he ML esiaor (dashed line) for The values of r are shown in he legend 0 0 σ =6006 db σ =00000 db σ =16478 db σ =91186 db σ =1938 db /σ [db] Fig 6: CRLB for and ^ for MÁ The plos appear idenical a he curren resoluion r is a paraeer ore convenien han he classical verical raining schee and provides he sae perforance We have also copared differen placeen schees using he CRLB for channel esiaion and he uual inforaion beween he inpu and he oupu γ [db] Fig 4: Horizonal placeen, CRLB vs r, l is a paraeer 5 CONCLUSIONS In his paper we have presened a channel esiaion approach for space-ie block codes The raining is done superiposed pilo sybols and he esiaion is ML seiblind We have inroduced he horizonal placeen of he raining sybols ha is 6 REFERENCES [1] AF Naguib, V Tarokh, N Seshadri and AR Calderbank, A space-ie coding ode for high-daa-rae wireless counicaions IEEE Journal on Seleced Areas in Counicaions, 16(8): ,1998 [] V Tarokh; H Jafarkhani and AR Calderbank, Space-ie block codes fro orhogonal designs IEEE Trans on Inforaion Theory, 45(5): , 1999 [3] TP Holden and K Feher, A spread specru based syse echnique for synchronizaion of digial obile cou-

5 nicaion syses IEEE Trans Broadcasing,Proceedings- Counicaions, 36(3): , 1990 [4] P Hoeher and F Tufvesson, Channel esiaion wih superiposed pilo sequence In GLOBECOM 99, pp , vol4, Rio de Janeiro, Brasil, Dec 1999 [5] JH Manon; IY Mareels and Y Hua, Affine precoders for reliable counicaions In ICASSP 00, Vol 5, pp , Isanbul, Turkey, June 00 [6] D Makrakis and K Feher, A novel pilo inserion-exracion echnique based on spread specru Affine precoders for reliable counicaions presened a Miai Technicon, Miai, 1987 [7] S Adireddy and L Tong, Opial Placeen of Known Sybols subied o IEEE Trans Inforaion Theory in July, 000 [8] S Adireddy and L Tong, Opial Ebedding of Known Sybols Proceedings of he 000 Conference on Inforaion Sciences and Syses (Princeon, NJ), March 000 [9] GGanesan and P Soica, Space-Tie Diversiy Signal Processing Advances in Wireless and Mobile Counicaions, Prenice Hall, 001, pp [10] S Ohno and G B Giannakis, Opial Training and Redundan Precoding for Block Transissions wih Applicaion o Wireless OFDM IEEE Transacions on Counicaions, subied Noveber 000